Chap 17 Transport phenomena and Fermi liquid theory

1. Dept of Phys M.C. Chang Chap 17 Transport phenomena and Fermi liquid theory • Boltzmann equation • Onsager reciprocal relation • Thermal electric phenomena • Seebeck, Peltier, Thomson… • Classical Hall effect, anomalous Hall effect • Theory of Fermi liquid • e-e interaction and Pauli exclusion principle • specific heat, effective mass • 1st sound and zero sound

2. k r Boltzmann equation • Distribution function: f(r,k,t) (“g” in Marder’s) Number of electrons within d3r and d3k around (r,k) at time t For example, • Evolution of the distribution function without collision the phase-space density does not change in the comoving frame Larger △r  smaller △k Phase space is incompressible (Liouville’s Theorem)

3. supplementary With collision (due to disorder… etc): Boltzmann eq. ～ Source/drain • Transition rate for an electron at k → k’: • Wk, k’ (calculated by Fermi golden rule) In a crowded space, one needs to consider occupancy and summation • On the other hand, the transition rate for an electron to be scattered into k Grosso, SSP, p.404 (for elastic scattering) see Marder, Sec 18.2 for more

4. supplementary k k’ θ (valid for uniform E, B, T fields) Only the component of k’ // k contribute to the integral(if W depends only on θ) Transport relaxation time Note: for inelastic scatterings, detailed balance requires

5. Relaxation time approximation Unperturbed equilibrium dist Relaxation (allows energy dependence) Density and temperature gradients are allowed • Boltzmann eq. Let’s call this “Chamber’s formulation”

6. 1st, consider a system with electric field and temperature gradient, but no magnetic field • Electric current density For steady perturbations Electrochemical “force” σ is conductivity tensor • Thermal current density κ is thermal conductivity tensor

7. Coefficients of transport (matrices) They are of the form Define energy-resolved conductivity One example of Onsager relation T=0, H.W.

8. More on the conductivity consider ▽T=0. Ohm’s law: Fick’s law for “diffusion current”: • For electron gas at low T, • At T=0, is an integral over the FS For isotropic diffusion, Einstein relation (for degenerate electron gas)

9. Alternative form of conductivity If τ a const. Optical effective mass = m* if the carriers are near the band bottom ～ free electron gas • Fourier’s law on thermal conduction Note: this would induce electric current. To remedy this, see Marder, p.496. (Wiedemann-Franz law for metals)

10. supplementary More on the Onsager reciprocal relation (1931) 互易關係 • Transport processes near equilibrium(linear transport regime) • “kinetic coefficient” L is symmetric: For example, if Ex drives a current Jy, then Ey will drive a current Jx. (Ohm’s Law) • The Onsager relationis a result of fluctuation-dissipation theorem, plus time reversal symmetry. • A specific example: the conductivity tensor of a crystal is symmetric, whatever the crystal symmetry is. (Fick’s Law) (Fourier’s Law) • They are of the form Thermodynamic “flow” Thermodynamic “force” Thermodynamic conjugate variables

11. supplementary Simultaneous irreversible processes For example, Note: For many transport processes near equilibrium, the entropy production is a product of flow and force ∴Entropy production ～ a thermodynamic potential Nature likes to stay at the lowest potential → minimum entropy production (only in the linear regime) Same symmetry relation applies to this larger α matrix → LT12=L21 • if force 1 (e.g., a temperature gradient) drives a flow 2 (diffusion current), then force 2 (density gradient) will drive a flow 1 (heat current) ! Precursors of the symmetry relation (D.G. Miller, J Stat Phys 1995) 1968 • Stokes (1851), anisotropic heat conduction • Kelvin (1854), thermoelectric effect • … http://www.ntnu.no/ub/spesialsamlingene/tekhist/tek5/eindex.htm

12. - • Thermoelectric coupling • Seebeck effect (1821) Seebeck found that a compass needle would be deflected by a closed loop formed by two metals joined in two places, with a temperature difference between the junctions. or • In the absence of electric current Longitudinal T gradient → electrochemical potential (in metals or semiconductors) • Seebeck coefficient (aka thermoelectric power) 熱電功率 typical values observed: a few μV / K (Bi: ~ 100 μV / K) 溫差導致電位差

13. (2) Peltier effect(1834) • In a bi-metallic circuit without T gradient, a current flow would induce a heat flow (in metals or semiconductors) Peltier found that the junctions of dissimilar metals were heated or cooled, depending upon the direction of electrical current. (If L11 and L21 commute) • In practical applications of Seebeck/Peltier, the figure of merit(dimensionless) is (Prob 7) High electric conductivity and low thermal conductivity is good. (>< Wiedemann-Franz law) • Bi2Te3 ZT ~0.6 at room temperature • If ZT~4, then thermoelectric refrigerators will be competitive with traditional refrigerators.

14. Thermoelectric cooling Peltier element: Bismuth Telluride (p/n type connected in series) advantages • Solid state heating and cooling – no liquids. CFC free. • Compact instrument • Fast response time for good temperature control

15. Seebeck vs Peltier

17. ρxy B Hall effect (1879): classical approach

18. B Hall effect: semiclassical approach Recall “Chamber’s formulation” (without density and T gradients) We can now only count on vk for magnetic effect ExB drift Assume E⊥B

19. Q1:How do we get the next order terms? Q2: What if the orbit is not closed? • For ωcτ>>1, the first term is zero If the open orbit is along the x-direction (in real space), then See Kitel, QTS, p.244 Q3: What about ωcτ<<1?

20. supplementary ρH slope=RN saturation RAHMS H Anomalous Hall effect(Edwin Hall, 1881): Hall effect in ferromagnetic (FM) materials FM material • Ingredients required for a successful theory: • magnetization (majority spin) • spin-orbit coupling(to couple the majority-spin direction to transverse orbital direction) The usual Lorentz force term Anomalous term

21. supplementary “Intrinsic” AHE due to the Berry curvature Mn5Ge3 Zeng et al PRL 2006 A transverse current To leading order, After averaging over long-wavelength spin fluctuations, the calculated anomalous Hall conductivity is roughly linear in M. The S.S. refers to skew scattering. • Skew scattering from an impurity transition rate

22. charge EF B ↑↓ y 0 L +++++++ B ↑↑↑↑↑↑↑ ↑↑↑↑ ↑ ↑ ↑ ↑ ↑ ↑ spin ↑↑↑↑↑↑↑ EF ↑↑↑↑ ↓ ↑ ↑↑↑↑ y ↑↑↑↑↑↑↑ 0 L • classical Hall effect • Lorentz force • anomalous Hall effect charge spin EF • Berry curvature (int) • Skew scattering (ext) ↑ ↓ y 0 L • spin Hall effect • Berry curvature (int) • Skew scattering (ext) No magnetic field required !

23. -yT Bz Jx Thermo-galvano-magnetic phenomena Onsager relations • Expand to first order in B, The effect of B on thermo-induced electric current Ohm Hall Nernst 1826 1879 1886 Onsager relations The effect of B on electric-induced thermo current Ettingshausen Fourier Leduc-Righi (Thermal Hall effect) 1886 1807 1887 Landau, ED of continuous media, p.101

24. Beyond thermo-galvano-magnetic phenomena Optical (O) • E-T: Thomson effect, Peltier/Seebeck effect • E-B: Hall effect, magneto-electric material • E-B-T: Nernst/Ettingshausen effect, Leduc-Righi effect • E-O, B-O: Kerr effect, Faraday effect, photovoltaic effect, photoelectric effect • E-M, B-M: piezoelectric effect/electrostriction, piezomagnetic effect/magnetostriction • M-O: photoelasticity • ... Thermal (T) Mechanical (M) Electric (E,P) Magnetic (B,M) N B E solid state refrigerator solid state sensor solid state motor, artificial muscle ... Landau and Lifshitz, Electrodynamics of continuous media Scheibner, 4 review articles in IRE Transations on component parts, 1961, 1962

25. TABLE 1-3  Physical and Chemical Transduction Principles. (from “Expanding the vision of sensor materials” 1995)

26. Landau’s theory of Fermi liquid

27. Why e-e interaction can usually be ignored in metals? Typically, 2 < U/K < 5 • Average e-e separation in a metal is about 2 A • Experiments find e mean free path about 10000 A (at 300K) At 1 K, it can move 10 cm without being scattered! Why? • A collision event: k1 k3 k2 k4 • Calculate the e-e scattering rate using Fermi’s golden rule: The summation is over all possible initial and final states that obey energy and momentum conservation Scattering amplitude

28. If |E2| < E1, then E3+E4 > 0 (let EF=0) • But since E1+E2 = E3+E4, 3 and 4 cannot be very far from the FS if 1 is close to the FS. • Let’s fix E1, and study possible initial and final states. 1 2 3 Pauli principle reduces available states for the following reasons: If the scattering amplitude |Vee|2is roughly of the same order for all k’s, then E1+E2=E3+E4;k1+k2=k3+k4 • 2 e’s inside the FS cannot scatter with each other (energy conservation + Pauli principle), at least one of them must be outside of the FS. Let electron 1 be outside the FS: • One e is “shallow” outside, the other is “deep” inside also cannot scatter with each other, since the “deep” e has nowhere to go.

29. (let the state of electron 1 be fixed) • number of initial states = (volume of E2 shell)/Δ3k • number of final states = (volume of E3 shell)/Δ3k • (E4 is uniquely determined) • τ-1 ～ V(E2)/Δ3k x V(E3)/Δ3k ← number of states for scatterings ∴τ-1 ～(4π/Δ3k)2 kF2|k2-kF|×kF2|k3-kF| Total number of states for particle 2 and 3 = [(4/3)πkF3/ Δ3k]2 • The fraction of states that “can” participate in the scatterings • = (9/kF2) |k2-kF|× |k3-kF| • ～(E1/EF)2 • Finite temperature: • ～(kT/EF)2 ～ 10-4 at room temperature (1951, V. Wessikopf) In general → e-e scattering rateT2 • need very low T (a few K) and very pure sample to eliminate thermal and impurity scatterings before the effect of e-e scattering can be observed.

30. 1962 • Landau’s theory of the Fermi liquid (1956) • Strongly interacting fermion system • → weakly interacting quasi-particle (QP) system • 1-1 correspondence between fermions and QPs (fermion, spin-1/2, charge -e). • adiabatic continuity:As we turn off the interaction, the QPs smoothly change back to noninteracting fermions. • The following analysis applies to a neutral, isotropic FL, such as He-3. ～ a particle plus its surrounding, finite life-time assumptions Q: Is this trivial? Another application: He-3 TF=7 K

31. is an effective interaction between QPs near FS Similarity and difference with free electron gas For a justification, see Marder • QP distribution (at eq.) ← if no other ext perturbations kF is not changed by interaction! • Due to external perturbations the distribution will deviate from the manybody ground state (no perturbation) at T=0 • Thermal • Non-thermal (T=0) (density perturbation, magnetic field… etc) • In general • QP energy In absence of other QPs • Total number

32. If there is no magnetic field, nor magnetic order, then is • independent of σ, and depends only on the relative spin directions. • Total energy This form is not good for charged FL (with long-range interaction) (forward scattering amplitude) For example, Quinn, p.384 (recall the Fock interaction in ch 9)

33. Fermi velocity • Effective mass Note: The use of follows Coleman’s note, Baym and Pethick etc, but not Marder’s. We don’t want these quantities to depend on perturbation • DOS See ch 7 • Specific heat (to lowest order) See ch 6 same as non-interacting result except for the effective mass.

34. C is proportional to T, but the slope gives an effective mass 103 times larger! ρ is proportional to T2, also a FL behavior (30 bar) Specific is linear in T below 20 mK Heavy fermion material (CeAl3) He-3 Giamarchi’s note, p.88

35. Effective mass of a QP (I) (total) “Particle” current (1) Z.Qian et al, PRL 93, 106601 (2004)

36. Effective mass of a QP (II) On the other hand, give particles an active boost (with p-h excitations) (2) (1)=(2) → (δf is arbitrary) If m* is spin-indep, (nonmagnetic FL) then (an integral over the FS) (Only for QPs near the Fermi surface) see Fradkin’s note, Pathria p.296

37. (a passive “boost”) Nozieres and Pines, p.37

38. k θ k’ • Introducing Fermi liquid parameters • Moments of over the FS provide the most important information about interactions (e.g., see the previous m* formula ) • let For spherical FS, ukk’ depends only on θ and decompose • Dimensionless parameters A small set of parameters for various phenomena For example, determined from specific heat. m*/m～3 for He-3 H.W.

39. More on the effective mass • recall Backflow correction (to ensure current conservation) diverges when (～ Mott transition)

40. Compressibility of Fermi liquid At fixed S or T (little difference near T=0) Note: Before compression At T=0, Both k and k’ lie on FS, and Ukk’ depends only on cosθ, ∴ Ak is indep. of k. Note: Slightly different from Marder’s (see Baym and Pethick, p.11) ≡ Ak (indep. of σ if not magnetized) = F0s

41. Dependence of various quantities on δμ • Note: • For attractive interaction, • If , then κT diverges, • and FS will become unstable to deformation (spontaneous breaking of rotational symmetry). • This is called Pomeranchuk instability (1958). For example, nematic FL. 向列型 0910.4166

42. Deformation of Fermi sphere and the FL parameters From Coleman’s note

43. κ compressibility summary For He-3, (larger effective mass) (more spin polarizable) (less compressible) From Coleman’s note

44. Travelling wave: firstly, 1st sound(i.e., the usual pressure wave) Velocity of the 1st sound F0S=10.8 for He-3, determined by measuring C1

45. γ • Zero sound(predicted by Landau, verified by Wheatley et al 1966) • usual sound requires ωτ<<1 (mean free path ℓ<<λ) • when ωτ→1 , sound is strongly absorbed • when ωτ>>1, sound propagation is again possible • zero sound is a collisionless sound ～ plasma wave in charged FL • no thermal equilibrium in each volume element • to get the zero sound, one can increase ω or decrease T (to increase τ) Can exist at 0 K Oscillation of Fermi sphere 1st sound t zero sound t (egg-like shape) Giamarchi’s note, p.102

46. Boltzmann-like approach (requires ) Instead of the semiclassical equations, one uses (collisionless) consider No r-dependence hidden in ε. (indep of spin) To order δf

47. (1) Assume a(θ) ～const. decompose • only F0s > 0 (repulsion) can have a solution • for

48. when QP velocity > C0 (s<1), the integral has a pole at , a QP would emit “supersonic” zero sound Analogies: • Supersonic shock wave • Cherenkov EM radiation from “superluminal” charged particles

49. vF Transition from the 1st sound to the zero sound Dispersion of zero sound in He-3 (from neutron scattering exp’t) Superfluid transition Vollhardt and Woelfle, p.45 Aldrich et al, PRL 1976

50. coherent In addition to collective excitations (zero sound, plasma), there are also particle-hole excitations incoherent particle-hole excitation: For charged FL only (more in ch 23) Q: what is the particle-hole band for 1-dim electron liquid? H. Godfrin et al, Nature 2012